From this I learned exactly where to put stuff and now I can get
perfect positioning of each element of fingering even inside a chord.
Here's the result of my experiment on the chord in Tom's sample code:

It looks kind of ugly in the code but it does allow excellent
placement of both the fingerings and the string indications. This is
definitely something I'll be using in my own scores. I just wish it
worked in variables (the docs on \tweak say it's a known issue that
\tweak doesn't work in a variable).

BTW this still didn't fix the problem Tom cited about the fingering
Orientation (down down up) being partially ignored. That's work for
another day...

Thanks for the tips, James and Trevor!
Jon
Trevor Daniels wrote:

Jonathan

Have a look at NR 5.3.4 The \tweak command. I rewrote this recently,
so I'd like to know whether or not this is clear and helpful. Thanks.

While it's possible to position fingerings very precisely using
extra-offset,
you can only do so to notes inside chord constructs. Fine. The
problem
is that if you're trying to position the fingerings for an actual
chord

(i.e. more than 1 note at the same time inside a chord construct),

isn't that what \tweak is for? I mean, the only time I see it is
when a specific change needs to be made inside a chord construct.
Again, I know nothing about what's going on here.